由有限值函数生成的巴拿赫空间：超自反刚性、汉克尔算子与普适性
Banach Spaces Generated by Finite-Valued Functions: Superreflexive Rigidity, Hankel Operators, and Universality
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中文总结 AI 辅助
该研究建立有限值函数生成巴拿赫空间的超自反性等几何性质与语言正则性的关联，构造出含所有可分实巴拿赫空间等距拷贝的二元语言空间，揭示正则语言的几何刚性与非正则情形的普适性差异。
中文摘要 AI 辅助
我们研究由取值于固定有限集合的一致有界函数族生成的巴拿赫空间，并建立了连接生成族基数与其闭线性张成几何的刚性原理。我们证明，当且仅当生成族有限时，这类空间是超自反的，等价于所得巴拿赫空间是有限维的。核心依据是一个有限范围的扩散模型障碍，表明在上确界范数下，无限有限值函数族无法生成超自反空间。该原理被应用于由形式语言左导数的特征函数生成的巴拿赫空间，它从有限维性、超自反性及等价一致凸范数的存在性角度，给出了正则语言的几何刻画。我们还得到了相关语言空间的典范表示，即其为紧移位轨道闭包上连续函数空间的坐标函数子空间。对应的语言汉克尔算子被证明仅对正则语言是紧的；在非正则情形下，我们确定了其与紧算子及有限秩算子的精确距离，并计算了所有非平凡逼近数。最后，我们构造了一个单一二元语言，其关联的巴拿赫空间包含每个可分实巴拿赫空间的等距拷贝。这些结果揭示了正则语言相关的几何刚性与非正则情形下可能出现的普适性之间的显著对比。
英文摘要
We investigate Banach spaces generated by uniformly bounded families of functions taking values in a fixed finite set and establish a rigidity principle connecting the cardinality of the generating family with the geometry of its closed linear span. We prove that such a space is superreflexive precisely when the generating family is finite, or equivalently, when the resulting Banach space is finite-dimensional. The main ingredient is a finite-range spreading-model obstruction showing that an infinite family of finite-valued functions cannot generate a superreflexive space under the supremum norm. This principle is applied to Banach spaces generated by the characteristic functions of the left derivatives of formal languages. It yields geometric characterizations of regular languages in terms of finite dimensionality, superreflexivity, and the existence of an equivalent uniformly convex norm. We also obtain a canonical representation of the associated language space as a coordinate-function subspace of a space of continuous functions on a compact shift-orbit closure. The corresponding language Hankel operator is shown to be compact exactly for regular languages. In the nonregular case, we determine its exact distance from both the compact and finite-rank operators and compute all its nontrivial approximation numbers. Finally, we construct a single binary language whose associated Banach space contains an isometric copy of every separable real Banach space. These results reveal a sharp contrast between the geometric rigidity associated with regular languages and the universality that may occur in the nonregular setting.
